# The action for a 5-brane to a string

1. Feb 22, 2013

### rbwang1225

Given an action for a 5-brance
$S=\int d^6x\left[\sqrt{-g}{1\over {4\partial_r a\partial^r a}}\partial_ma(x)F^{*mnl}F_{nlp}\partial^pa(x)+ \sqrt{-\det(g_{mn}+i\tilde F_{mn})}\right],$
where $x^m$ $(m=0,1...,5)$ are local coordinates of the worldvolume, $g_{mn}(x)=\partial_mX^M(x)g_{MN}\partial_nX^N(x)$ is a worldvolume metric
induced by embedding into curved target space with the metric $g_{MN}(X)$ parametrized by coordinates $X^M$ $(M,N=0,...,D-1)$; $F_{mnl}=2(\partial_{l}A_{mn}+ \partial_{m}A_{nl}+\partial_{n}A_{lm})$ is the field strength of an antisymmetric worldvolume gauge field $A_{mn}(x)$;
$g=\det{g_{mn}}$; $F^{*lmn}$ is the dual field strength:
$$F^{*lmn}={1\over {6\sqrt{-g}}} \varepsilon^{lmnpqr}F_{pqr}$$
and
$$\tilde F_{mn}\equiv{1\over{\sqrt{(\partial a)^2}}}F^{*}_{mnl}\partial^la(x).$$

The scalar field $a(x)$ ensures manifest $d=6$ covariance of the model and is completely auxiliary.

My question is how do I reduce it to a 1-brance, i.e., a string?
Is there any standard procedure for this kind of problem?

Any devices would be very appreciated.

2. Feb 26, 2013

### mitchell porter

A string comes from compactifying an M-theory 2-brane on a circle. The M-theory 5-brane doesn't become a string, it becomes a D5-brane (or D4-brane) or a "heterotic fivebrane". Although the 5-brane and the 2-brane are related, and there may be some compactification where a wrapped 5-brane is dual to a string.